Related papers: The Noether numbers for cyclic groups of prime ord…

A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that…

Based on recent successes concerning permutation resolutions of representations by Balmer and Gallauer we define a new invariant of finite groups: the p-permutation dimension. We define this analogously to the global dimension of a ring by…

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…

Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and…

We examine the assumptions behind Noether's theorem connecting symmetries and conservation laws. To compare classical and quantum versions of this theorem, we take an algebraic approach. In both classical and quantum mechanics, observables…

Multiplicative order of an element $a$ of group $G$ is the least positive integer $n$ such that $a^n=e$, where $e$ is the identity element of $G$. If the order of an element is equal to $|G|$, it is called generator or primitive root. This…

Consider a Noetherian domain $R$ and a finite group $G \subseteq Gl_n(R)$. We prove that if the ring of invariants $R[x_1, \ldots, x_n]^G$ is a Cohen-Macaulay ring, then it is generated as an $R$-algebra by elements of degree at most…

We prove a fractional Noether's theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula…

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$.…

In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds…

A group $G$ is said to be equationally Noetherian if every system of equations in $G$ is equivalent to a finite subsystem. We show that all free-by-cyclic groups are equationally Noetherian. As a corollary, we deduce that the set of…

The prime graph (or Gruenberg-Kegel graph) of a finite group $G$ is a familiar graph. In this paper first, we investigate the structure of the finite groups with a non-complete prime graph. Then we prove that every alternating group…

We use methods of combinatorial number theory to prove that, for each $n>1$ and any prime $p$, some homotopy group $\pi_i(SU(n))$ contains an element of order $p^{n-1+ord_p([n/p]!)}$, where $ord_p(m)$ denotes the largest integer $\alpha$…

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following Theorem A: Let…

For any polynomial mapping $F=(F_1,\dots ,F_n)$ of $\Cal C^n$ with a finite number of zeros we define the Noether exponent $\nu(F)$. We prove the Jacobi formula for all polynomials of degree strictly less than $\sum_{i=1}^n (\deg…

A classical theorem of Forster asserts that a finite module $M$ of rank $\leq n$ over a Noetherian ring of Krull dimension $d$ can be generated by $n + d$ elements. We prove a generalization of this result, with "module" replaced by…

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g\,|\,g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $k(G)=k(x_g\,|\,g\in G)^G$ is…

We define $g(n)$ to be the maximal order of an element of the symmetric group on $n$ elements. Results about the prime factorization of $g(n)$ allow a reduction of the upper bound on the largest prime divisor of $g(n)$ to $1.328\sqrt{n\log…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…